The fact that the \(\mathfrak {Fr}\)  is B-UBP is just the classical Banach-Steinhaus theorem. On the other hand, many classical filters \(\mathfrak {F}\)  do not enjoy this property, because of the existence of \(\mathfrak {F}\) -unbounded pointwise \(\mathfrak {F}\) -convergent sequences in dual Banach spaces. The latter effect was remarked in
[1]} for the statistical convergence and was investigated in detail in [2]}, [3]}, and [4]}.
